It was May 2013, and Cook, the chief executive of Apple, appeared before a Senate investigative subcommittee. After a lengthy inquiry, it found that the company had avoided tens of billions of dollars in taxes by shifting profits into Irish subsidiaries that the panel’s chairman called “ghost companies.”

“We pay all the taxes we owe, every single dollar,” Cook declared at the hearing. “We don’t depend on tax gimmicks,” he went on. “We don’t stash money on some Caribbean island.”

True enough. The island Apple would soon rely on was in the English Channel.

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Five months after Cook’s testimony, Irish officials began to crack down on the tax structure Apple had exploited. So the iPhone maker went hunting for another place to park its profits, newly leaked records show. With help from law firms that specialize in offshore tax shelters, the company canvassed multiple jurisdictions before settling on the small island of Jersey, which typically does not tax corporate income.

Apple has accumulated more than $128 billion in profits offshore, and probably much more, that is untaxed by the United States and hardly touched by any other country. Nearly all of that was generated over the past decade.

The previously undisclosed story of Apple’s search for a new island tax haven and its use of Jersey is among the revelations emerging from a cache of secret corporate records from Appleby, a Bermuda-based law firm that caters to businesses and the wealthy elite.

The records, shared by the International Consortium of Investigative Journalists with The New York Times and other media partners, were obtained by the German newspaper Süddeutsche Zeitung.

The documents reveal how big law firms help clients weave their way through the gaps between different countries’ tax rules. Appleby clients have transferred trademarks, patent rights, and other valuable intangible assets into offshore shell companies, avoiding billions of dollars in taxes. The rights to Nike’s Swoosh trademark, Uber’s taxi-hailing app, Allergan’s Botox patents, and Facebook’s social media technology have all resided in shell companies that listed as their headquarters Appleby offices in Bermuda and Grand Cayman, the records show.

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“US multinational firms are the global grandmasters of tax avoidance schemes that deplete not just US tax collection but the tax collection of most every large economy in the world,” said Edward Kleinbard, a former corporate tax adviser to such companies, now a law professor at the University of Southern California.

Indeed, tax strategies like the ones used by Apple — as well as Amazon, Google, Starbucks, and others — cost governments around the world as much as $240 billion a year in lost revenue, according to a 2015 estimate by the Organization for Economic Cooperation and Development. The group is overseeing efforts to change the global rules that permit such maneuvers.

Apple has found a tax haven in Jersey, leaving billions of dollars untouched by the US.(Andrew Testa/New York Times)

The disclosures come after last week’s proposals by Republican lawmakers to provide several new tax benefits for multinational companies, including cutting the federal corporate income tax rate from 35 percent to 20 percent. President Trump has said that US businesses are getting a bad deal under current rules.

But the documents show how major US companies find creative ways to avoid paying anything close to 35 percent.

Apple, for example, pays taxes at a small fraction of that rate on its offshore profits, according to calculations by the Times based on the company’s securities filings. Apple reports that nearly 70 percent of its worldwide profits are earned offshore.

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An Apple spokesman, Josh Rosenstock, declined to answer most questions about the company’s tax strategy. He did say that Apple had told regulators — in the United States and Ireland and at the European Commission — about the reorganization of its Irish subsidiaries. “The changes we made did not reduce our tax payments in any country,” he said.

He added: “At Apple we follow the laws, and if the system changes we will comply. We strongly support efforts from the global community toward comprehensive international tax reform and a far simpler system.”

Congressional Republicans are also seeking to impose a 10 percent tax on some of the profits that US businesses say are earned offshore — half the rate they are proposing for profits in the United States. The lawmakers have also proposed another break, permitting multinationals to bring home more than $2.6 trillion stowed offshore at sharply reduced tax rates. Both proposals, critics say, would only create additional incentives for businesses such as Apple to shift more profits into island hideaways.

Appleby is a member of the global network of lawyers, accountants, and bankers who set up or manage offshore companies and accounts for clients who want to avoid taxes or keep their finances a secret from authorities, business partners, or even spouses. The firm did not respond to questions from the Times about its work for Apple or other companies.

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Since the mid-1990s, multinationals based in the United States have increasingly shifted profits into offshore tax havens. Indeed, a tiny handful of jurisdictions — mostly Bermuda, Ireland, Luxembourg, and the Netherlands — now account for 63 percent of all profits that US multinational companies claim to earn overseas, according to an analysis by Gabriel Zucman, an assistant professor of economics at the University of California Berkeley. Those destinations hold far less than 1 percent of the world’s population.

Criticism of such profit-shifting was largely ignored until government finances around the globe came under pressure in the years following the 2008 financial crisis, when the practice led to government inquiries, tax inspector raids, media scrutiny, and promises of reform.

In May 2013, the Senate’s investigative subcommittee released a 142-page report on Apple’s tax avoidance, finding that the company was attributing billions of dollars in profits each year to three Irish subsidiaries that declared “tax residency” nowhere in the world.

Under Irish law, if a company can convince Irish tax authorities that it is “managed and controlled” abroad, it can largely escape Irish income tax. By seeming to run its Irish subsidiaries from its world headquarters in California, Apple ensured that Irish tax residency was avoided.

At the same time, US law dictated that the subsidiaries were only tax residents in the United States if incorporated there. The federal government permits taxes on any income generated by foreign units to be deferred indefinitely, as long as the company says those profits stay offshore.

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“Apple has sought the Holy Grail of tax avoidance: offshore corporations that it argues are not, for tax purposes, resident anywhere in any nation,” then-Senator Carl Levin, Democrat of Michigan who was the subcommittee chairman, said at the 2013 hearing.

Apple’s hunt for a tax haven is a familiar tale, said Reuven Avi-Yonah, director of the international tax program at the University of Michigan Law School, who also reviewed the Appleby documents.

“This is how it usually works: You close one tax shelter, and something else opens up,” he said. “It just goes on endlessly.”